State the parallel axis theorem.

Short Answer:

The Parallel Axis Theorem states that the moment of inertia of a body about any axis parallel to its centroidal axis is equal to the sum of the moment of inertia about the centroidal axis and the product of the mass (or area) of the body and the square of the perpendicular distance between the two axes.
Mathematically, it is expressed as:

(for area moment of inertia) or

(for mass moment of inertia),
where is the moment of inertia about the centroidal axis and is the distance between the two axes.

Detailed Explanation :

Parallel Axis Theorem

In Engineering Mechanics, the Parallel Axis Theorem is a very important principle used to determine the moment of inertia of a body or an area about an axis that does not pass through its centroid. Since most engineering components and structures are not always rotated or bent about their centroidal axes, this theorem allows engineers to calculate the moment of inertia about any other axis parallel to the centroidal one.

The theorem applies to both mass moment of inertia (for solid bodies in rotation) and area moment of inertia (for plane figures such as beams and plates).

Statement of the Theorem

The Parallel Axis Theorem can be stated as:
“The moment of inertia of a body about any axis parallel to the centroidal axis is equal to the sum of the moment of inertia about the centroidal axis and the product of the mass (or area) of the body and the square of the perpendicular distance between the two axes.”

Mathematically:

  1. For area moment of inertia,
  1. For mass moment of inertia,

where,

  • = moment of inertia about the new axis,
  • = moment of inertia about the centroidal axis,
  • = area of the body (for area moment of inertia),
  • = mass of the body (for mass moment of inertia),
  • = perpendicular distance between the centroidal axis and the parallel axis.

Derivation of the Parallel Axis Theorem

Let us derive the theorem for a plane area.

Consider an area having a centroidal axis and another axis , parallel to it at a distance .
Let be a small elemental area located at a distance from the centroidal axis.

Now, the distance of the element from the new axis is .

The moment of inertia of the area about the new axis is given by:

Expanding the term,

 

Now,

and since the centroid is the average position of the area,

(because the positive and negative moments balance each other about the centroid).

Also,

(subtotal of all the elemental areas equals total area).

Substitute these values into the equation:

Hence,

This is the Parallel Axis Theorem for Area Moment of Inertia.

Similarly, for mass moment of inertia:

Physical Meaning

The theorem explains how the distribution of area or mass relative to different axes affects the moment of inertia.

  • The centroidal axis gives the minimum moment of inertia, as it passes through the center of the body’s area or mass distribution.
  • Any other axis parallel to it will have a greater moment of inertia, because of the added term or , which accounts for the shift of the axis away from the centroid.

In simple terms, moving the axis farther from the centroid increases the body’s resistance to bending or rotation.

Applications of the Parallel Axis Theorem

The theorem is widely used in both mechanical and civil engineering applications. Some important uses are:

  1. Calculating Moment of Inertia of Composite Sections:
    It is used to find the moment of inertia of complex shapes (like T-sections, I-beams, and L-sections) by splitting them into simple parts and shifting each part’s moment of inertia to a common reference axis.
  2. Structural Engineering:
    Helps determine bending stresses and deflections in beams when the neutral axis does not pass through the centroid of the section.
  3. Rotational Motion of Rigid Bodies:
    In dynamics, the theorem helps find the moment of inertia of rotating parts such as flywheels, pulleys, and gears about their rotation axes.
  4. Machine Design:
    Used to calculate the polar moment of inertia of shafts and rotating components for strength and stability analysis.
  5. Analysis of Stability:
    In stability problems, such as rolling motion or balancing rotating bodies, it helps determine how shifting mass affects inertia and balance.

Example Problem

Example:
Find the moment of inertia of a rectangular section about a base axis, given:

  • Breadth ,
  • Height .

Solution:
Moment of inertia about the centroidal axis (I_G):

Distance between centroidal axis and base = .
Using the Parallel Axis Theorem,

where .

Substitute values:

Hence, the moment of inertia about the base is .

Key Points

  • The theorem applies only to parallel axes.
  • The centroidal axis provides the minimum moment of inertia.
  • The term or increases with the square of the distance between the axes.
  • It is valid for both plane areas and solid bodies.
Conclusion

In conclusion, the Parallel Axis Theorem helps determine the moment of inertia of a body about any axis parallel to its centroidal axis. It shows that the M.I. increases as the axis moves away from the centroid due to the added or term. This theorem is widely used in structural, mechanical, and dynamic systems to analyze bending, rotation, and balance. Understanding and applying this principle is essential in designing safe and efficient mechanical structures.